# 1994 AHSME Problems/Problem 2

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## Problem

A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle? $[asy] draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); draw((0,5)--(10,5)); draw((3,0)--(3,7)); label("6", (1.5,6)); label("?", (1.5,2.5)); label("14", (6.5,6)); label("35", (6.5,2.5)); [/asy]$

$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 25$

## Solution

[asy] pair A=(0,0),B=(10,0),C=(10,7),D=(0,7),EE=(0,5),F=(10,5),G=(3,0),H=(3,7); path BG=shift(0,-0.5)*(B--G); path BF=shift(0.5,0)*(B--F); path FC=shift(0.5,0)*(F--C); path DH=shift(0,0.5)*(D--H); draw(A--B--C--D--cycle); draw(EE--F); draw(G--H); draw(BG,L=Label("7",position=MidPoint,align=(0,-1)),arrow=Arrows(),bar=Bars,red); draw(BF,L=Label("5",position=MidPoint,align=(1,0)),arrow=Arrows(),bar=Bars,red); draw(FC,L=Label("2",position=MidPoint,align=(1,0)),arrow=Arrows(),bar=Bars,red); draw(DH,L=Label("3",position=MidPoint,align=(0,1)),arrow=Arrows(),bar=Bars,red); label("6", (1.5,6)); label("15", (1.5,2.5),blue); label("14", (6.5,6)); label("35", (6.5,2.5)); [/asy]

We can easily see the dimensions of each small rectangle. So the area of the last rectangle is $3\times 5=\boxed{\textbf{(B) }15}$.

## Solution 2

[asy] pair A=(0,0),B=(10,0),C=(10,7),D=(0,7),EE=(0,5),F=(10,5),G=(3,0),H=(3,7); path CH=C--H; path BF=B--F; path FC=F--C; path DH=D--H; draw(A--B--C--D--cycle); draw(EE--F); draw(G--H); draw(CH,L=Label("b",position=MidPoint,align=(0,1))); draw(BF,L=Label("y",position=MidPoint,align=(1,0))); draw(FC,L=Label("x",position=MidPoint,align=(1,0))); draw(DH,L=Label("a",position=MidPoint,align=(0,1))); label("6", (1.5,6)); label("14", (6.5,6)); label("35", (6.5,2.5)); [/asy]

In case its not immediately obvious from inspection what the dimensions of the small rectangles are, we can work it out. We know $ax$ and $bx$ and $by$ and we want to know $ay$. We can compute it as follows $ay = \frac{(ax)(by)}{bx} = \frac{ 6\cdot 35 }{14} = 15$ and the answer is $\fbox{B}$